# Fungrim entry: 21f4f9

$\frac{d^{n}}{{d z}^{n}} \left[\psi^{(m)}\!\left(z\right)\right] = \psi^{(m + n)}\!\left(z\right)$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}$
TeX:
\frac{d^{n}}{{d z}^{n}} \left[\psi^{(m)}\!\left(z\right)\right] = \psi^{(m + n)}\!\left(z\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}
Definitions:
Fungrim symbol Notation Short description
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
DigammaFunction$\psi\!\left(z\right)$ Digamma function
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Source code for this entry:
Entry(ID("21f4f9"),
Formula(Equal(ComplexDerivative(Brackets(DigammaFunction(z, m)), For(z, z, n)), DigammaFunction(z, Add(m, n)))),
Variables(n, m, z),
Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)), Element(z, CC), NotElement(z, ZZLessEqual(0)))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC